CAPÍTULO V.- CONCLUSIONES Y RECOMENDACIONES
ANEXO 3. Cuestionario de encuesta
A number of other cases may be considered when it comes to exact derivations of the sandwich co-
variance matrix. For instance, there is some ambiguity in how observations are grouped in the com-
posite marginal blockwise likelihood, and the sequence of observations in the composite conditional
K-sequential neighbours likelihood. In the case of the marginal blockwise likelihood, a block could be formed by starting from the observation indexed byq∈ {1,2, ...,W}and taking everyB-th observa- tion thereafter on the number line. Given the poor performance of the composite conditionalK-nearest neighbours likelihood under infill, we could also consider a composite likelihood that only includes ev-
ery second conditional density; that is,
L
C(θ;y) =∏ dN−12 e
i=1 f(y(s2i)|y(s2i−1),y(s2i+1);θ). This is more in line with the original construction of Besag (1974). Finally, the composite conditional and composite
marginal pairwise likelihoods are also of interest; though since these likelihood functions incorporate all
pairs of observations, a potential complication is that the composition matrix will not be sparse.
Theoretical results for maximum composite likelihood estimation could also be obtained in a two-
dimensional exponential covariance setting on an equally-spaced lattice. Results under the full like-
lihood case whereC(h) =σ2exp(−α1|h1| −α2|h2|))for a separation vectorh= (h1,h2)T are available due to Ying (1993). In particular, the separation of the covariance structure by direction allows the
covariance matrix to be expressed as a Kronecker product of exponential covariance matrices from the
one-dimensional case. This case is especially notable due to the parameters being consistent and asymp-
totically normal under infill.
The two-dimensional case also has implications in the context of spatio-temporal data, where the tem-
poral dimension has a naturally expanding domain. Note that the aforementioned covariance struc-
ture can be slightly restructured to the separable and stationary spatio-temporal covariance structure
C(h,t) =C(h)C(t) =σ2exp(−α1|h|)exp(−α2|t|). Thus, it would also be interesting to consider whether we have consistency inα1 if the spatial lattice scheme remains fixed and finite at all time points, and
only the temporal domain expands.
It would also be worth considering other stationary covariance models that are used in practice, such
as the Mat´ern covariance structure or non-separable models. In these cases, an important consideration
would be whether the inverse of the covariance matrix is attainable in a closed-form, or at least a sparse
Detailed derivation of the trace of a four-
matrix product
We will present a detailed derivation of tr(MΣMΣ)from (3.20) in Section 3.3.2, where
M= 1 1−ρ2 " 1+ ρ 2 1+ρ2 I+2ρ2A+−ρ2+ ρ 2 1+ρ2 B−2ρC+ ρ 2 1+ρ2D # = 1 1−ρ4[(1+2ρ 2)I+2( ρ2+ρ4)A−ρ4B−2(ρ+ρ3)C+ρ2D],
as per (3.13). Using the individual traces of four-matrix products from (3.19), we have that
tr(MΣMΣ) = 1 (1−ρ4)2[(1+4ρ 2+4ρ4)tr(IΣIΣ) +4ρ4(1+2ρ2+ ρ4)tr(AΣAΣ) +ρ8tr(BΣBΣ) +4ρ2(1+2ρ2+ρ4)tr(CΣCΣ) +ρ4tr(DΣDΣ) +4ρ2(1+3ρ2+2ρ4)tr(IΣAΣ)−2ρ4(1+2ρ2)tr(IΣBΣ) −4ρ(1+3ρ2+2ρ4)tr(IΣCΣ) +2ρ2(1+2ρ2)tr(IΣDΣ) −4ρ6(1+ρ2)tr(AΣBΣ)−8ρ3(1+2ρ2+ρ4)tr(AΣCΣ) +4ρ4(1+ρ2)tr(AΣDΣ) +4ρ5(1+ρ2)tr(BΣCΣ) −2ρ6tr(BΣDΣ)−4ρ3(1+ρ2)tr(CΣDΣ)] = σ 4 (1−ρ4)2[Q1+Q2],
where Q1=(1+4ρ2+4ρ4)DF+4ρ4(1+2ρ2+ρ4)(DF−2) +ρ8(DF−4) +8ρ2(1+3ρ2+3ρ4+ρ6)(DF−1) +2ρ4(1+ρ4)(DF−2) +4ρ6(1+ρ2)(DF−3) +4ρ2(1+3ρ2+2ρ4)(DF−2)−2ρ4(1+4ρ2+4ρ4)(DF−4) −16ρ2(1+3ρ2+2ρ4)(DF−1) +4ρ4(1+2ρ2)(DF−2) −4ρ6(1+ρ2)(DF−4)−32ρ4(1+2ρ2+ρ4)(DF−2) +8ρ6(1+ρ2)(DF−2) +16ρ6(1+ρ2)(DF−4) −4ρ8(DF−4)−16ρ4(1+2ρ2+ρ4)(DF−2) =(1+4ρ2+4ρ4)DF+ (−8ρ2−24ρ4−8ρ6+8ρ8)(DF−1) + (4ρ2−26ρ4−64ρ6−34ρ8)(DF−2) + (4ρ6+4ρ8)(DF−3) + (−2ρ4+4ρ6+ρ8)(DF−4) =(1−48ρ4−64ρ6−21ρ8)DF+84ρ4+108ρ6+44ρ8.
By repeatedly applyingun=unρ+21−n, we also have
Q2=2[(1+4ρ2+4ρ4)uDF+4ρ4(1+2ρ2+ρ4)uDF−2+ρ8uDF−4 +16ρ2(1+2ρ2+ρ4)uDF−1+4ρ6uDF−3 +4ρ2(1+3ρ2+2ρ4)uDF−1−2ρ6(1+2ρ2)uDF−3 −8ρ2(1+3ρ2+2ρ4)uDF−1+4ρ2(1+2ρ2)uDF−1 −4ρ6(1+ρ2)uDF−3−16ρ4(1+2ρ2+ρ4)uDF−2 +8ρ4(1+ρ2)uDF−2+8ρ6(1+ρ2)uDF−3 −4ρ6uDF−3−16ρ4(1+ρ2)uDF−2] =2[(1+4ρ2+4ρ4)uDF+2ρ2(8+14ρ2+4ρ4)uDF−1 −4ρ4(5+8ρ2+3ρ4)uDF−2+2ρ6uDF−3+ρ8uDF−4]
=2[(1+4ρ2+4ρ4)uDF+2ρ2(8+14ρ2+4ρ4)uDF−1 −4ρ4(5+8ρ2+3ρ4)uDF−2+3ρ6uDF−3−ρ8(DF−4)] =2[(1+4ρ2+4ρ4)uDF+2ρ2(8+14ρ2+4ρ4)uDF−1 −ρ4(17+32ρ2+12ρ4)uDF−2−3ρ6(DF−3)−ρ8(DF−4)] =2[(1+4ρ2+4ρ4)uDF−ρ2(1+4ρ2+4ρ4)uDF−1 +ρ4(17+32ρ2+12ρ4)(DF−2)−3ρ6(DF−3)−ρ8(DF−4)] =2[ρ2(1+4ρ2+4ρ4)(DF−1) +ρ4(17+32ρ2+12ρ4)(DF−2)−3ρ6(DF−3)−ρ8(DF−4)] =2[(ρ2+21ρ4+33ρ6+11ρ8)DF−ρ2−38ρ4−59ρ6−20ρ8]. Thus, tr(MΣMΣ) = σ 4 (1−ρ4)2[Q1+Q2] = σ 4 (1−ρ4)2[(1+2ρ 2−6ρ4+2ρ6+ ρ8)DF−2ρ2+8ρ4−10ρ6+4ρ8] = σ 4 (1−ρ4)2[(1−ρ 2)2(1+4ρ2+ ρ4)DF−2ρ2(1−ρ2)2(1−2ρ2)] = σ 4 (1+ρ2)2[(1+4ρ 2+ ρ4)DF−2ρ2+4ρ4], as required.
This procedure can be repeated to obtain expressions for tr(MΣM0Σ)and tr(M0ΣM0Σ). Similarly, we
can use this method to find the traces of the four-matrix products in (3.34) for the composite marginal
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